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Even Perfect Number

An even perfect number is an even positive integer n whose positive proper divisors sum to n. Equivalently, its divisor function satisfies sigma(n)=2n.

The Euclid-Euler theorem states that the even perfect numbers are exactly the numbers 2^(p-1)(2^p-1) for which 2^p-1 is a Mersenne prime (Dickson 2005, p. 19).

All known perfect numbers are even, and Ochem and Rao (2012) have shown that any odd perfect number must be larger than 10^(1500).

See also

Euclid-Euler Theorem, Mersenne Prime, Odd Perfect Number, Perfect Number

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References

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 19, 2005.Ochem, P. and Rao, M. "Odd Perfect Numbers Are Greater than 10^(1500)." Math. Comput. 81, 1869-1877, 2012.

Referenced on Wolfram|Alpha

Even Perfect Number

Cite this as:

Weisstein, Eric W. "Even Perfect Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EvenPerfectNumber.html

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